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Full text of "Treatise On Analysis Vol-Ii"

214       XIII    INTEGRATION

belongs to some Un, which intersects only finitely many of the Vfc), and we

have |/*|*(/) =+oo, because M*(/)   |/i|(/i) for all /i.

fc= i

If S = Supp(/*) is compact,  the first part of the proof shows that
\fj\(f)  M(S) ' sup |/(jc)| for all functions/e #C(X), and therefore ju is a

xeS

continuous function on the Frechet space #C(X). Conversely, let A be a con-
tinuous linear form on this space, so that there exists a compact set K a X and
a constant c> 0 such that |A(/)| ^ c  sup |/(jc)| for all/s C(X) (12.14.6). If

xeK

L is any compact subset of X, we then have |A(/)| ^ c  ||/|| for all
/e Jfc(X; L), and hence the restriction u of A to Jf C(X) is a measure on X.
Moreover, if A is a continuous mapping of X into [0,1], with compact
support and such that h(x) = 1 for all xeK ((3.18.2) and (4.5.2)), then
MJh) = A(/) for all functions/e VfX)9 because/-//* is zero on K. From this
it follows immediately that the support of \JL is contained in Supp(/z); hence
every/e *C(X) is /i-integrable, and n(f) = j*(fK) = l(fh) = MJ). Q.E.D.

(13.19.4) If 7i: X->X' is a homeomorphism and fi is a measure on X, we
have Supp(7i(/0) = 7r(Supp(//)). This follows directly from the definitions.

PROBLEM

Let /x be a positive measure on X, and A a /n-measurable set. Let i(A) be the set of
points xeX such that there exists a compact neighborhood V of x in X for which
py n (X - A)) = 0. Show that /(A) is open and that /u(/(A) n (X - A)) = 0. (Con-
x-A  ft).)

20. BOUNDED MEASURES

For every (complex) measure /* on X, we define
(13.20.1)                       Ml-         sup

which is a finite real number, or + oo.

(1 3.20.2)    We have ||/i|| = |/*|*(1). Hence \\IJL\\ is finite if and only if the positive
measure |/t| is bounded (13.9), and in that case

total mass ofX. with respect to /.ce (Vn) of relatively compact open sets, as follows: